2013 can be written as

What is the maximum possible value of n for which 2013 can be written as a sum of n
consecutive positive integers?

value of m + n

Let m be the smallest odd positive integer for which (1+ 2 +......... + m) is a square of an integer
and let n be the smallest even positive integer for which (1 + 2 + ............ + n) is a square of an
integer. What is the value of m + n?

maximum possible number of distinct colours used

To each element of the set S = {1, 2,............, 1000} a colour is assigned. Suppose that for any
two elements  m & n of S, if 15 divides (m + n) then they are both assigned the same colour.
What is the maximum possible number of distinct colours used?

Pixie and Vedansi together have m marbles

Let Pixie and Vedansi together have m marbles, where m > 0.
Pixie says to Vedansi, "If I give you some marbles then you will have twice as many marbles
as I will have."
Vedansi replies to Pixie, "If I give you some marbles then you will have thrice as many marbles
 as I will have."
What is the minimum possible value of m for which the above statements are true?

S(5N + 2013)

Let S(M) denote the sum of the digits of a positive integer M written in base 10. Let N be
the smallest positive integer such that S(N) = 2013. What is the value of S(5N + 2013)?

red, green & blue

There are (p -1) red balls, p green balls and (p +1) blue balls in a bag. The number of ways of
choosing two balls from the bag that have di fferent colors is 299. What is the value of p?

no straight line can be drawn

Prove that no straight line can be drawn within a triangle which is greater than the greatest side.

Let N be the greatest integer multiple of 8

Let N be the greatest integer multiple of 8, no two of whose digits are the same. What is the remainder
when N is divided by 1000?

The product N of three positive integer

The product N of three positive integers is 6 times their sum and one of the integers is the sum of
the other two. find the sum of all possible values of N.

Find the 8th term of the sequence 1440, 1716, 1848

Find the 8th term of the sequence 1440, 1716, 1848, ...... whose terms are formed by multiplying the corresponding terms of two arithmetic sequences. 

In a triangle ABC, let G be the centroid

In a triangle ABC, let G be the centroid and let M, N be points in the interiors of the segments AB, AC
respectively, such that M, G, N are collinear. If r be the ratio of the triangle AMN to the area of the triangle
ABC, then evaluate r. 

Let ABC be an acute angled triangle

Let ABC be an acute angled triangle and let D be the midpoint of BC. If AB = AD, then find the value of
tanA/tanC

Consider a triangle ABC in the XY plane

Consider a triangle ABC in the XY plane with vertices A = (0,0), B = (1,1) & C = (9,1). If the line x = k
divides the triangle into two parts of equal area, find k;

the least possible value of (m + n)

If m, n are positive real numbers such that the lines mx + 9y = 5 and 4x + ny = 3 are parallel, find the least possible value of (m + n).

--7 to + 7

 How many ways can the integers from -7 to + 7  inclusive be arranged in a sequence such that the
absolute value of the numbers in the sequence does not decrease?

A line in the plane is called strange

A line in the plane is called strange if it passes through (a, 0) & (0, 10 - a)  for some a
in the interval [0,10].  A point in the plane is called charming if it lies in the fi rst quadrant
and also lies below some strange line. What is the area of the set of all charming points?

the differentiable function F

The diff erentiable function F : R to R  satisfi es F(0) = -1 and
d/dx ( F(x) ) = = sin(sin(sin(sin(x)))) .cos(sin(sin(x))).cos(sin(x)). cos(x)
Find F(x) as a function of x.

f be a differentiable

Let f be a diff erentiable real-valued function de fined on the positive real numbers.
The tangent lines to the graph of 'f ' always meet the y-axis 1 unit lower than where
they meet the function. If f(1) = 0, what is f(2)?

10 parabolas

Into how many regions can a circle be cut by 10 parabolas?

a sum of distinct factorials

How many positive integers less than or equal to 240 can be expressed as a sum of distinct factorials?
Consider 0! and 1! to be distinct.

monic cubic polynomial

Let p be a monic cubic polynomial such that p(0) = 1 and such that all the zeros of p'(x) are also
zeros of p(x). Find p. Note: monic means that the leading coeffi cient is 1.

degree exactly 5

How many polynomials of degree exactly 5 with real coeffi cients send the set {1, 2, 3, 4, 5, 6} to a
permutation of itself?

30 desks arranged in 5 rows of 6

A classroom has 30 students and 30 desks arranged in 5 rows of 6. If the class has 15 boys and
15 girls, in how many ways can the students be placed in the chairs such that no boy is sitting in front
of, behind, or next to another boy, and no girl is sitting in front of, behind, or next to another girl?

a * b = ab + a + b

Let a * b = ab + a + b for all integers a and b. Evaluate 1 * (2 * (3 * (4 * .....(99 * 100).....))).

a walk on a rectangular grid

 PIXIE takes a walk on a rectangular grid of n rows and 3 columns, starting from the bottom left corner.
At each step, he can either move one square to the right or simultaneously move one square to the left
and one square up. In how many ways can he reach the center square of the topmost row?

drawn centered at the origin

A circle of radius 6 is drawn centered at the origin. How many squares of side length 1 and integer
coordinate vertices intersect the interior of this circle?

{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

How many ordered pairs (S, T) of subsets of {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} are there whose union contains
exactly three elements?

Dragons take up 1 x 1 squares

Dragons take up 1 x 1 squares in the plane with sides parallel to the coordinate axes such that
the interiors of the squares do not intersect. A dragon can re at another dragoon if the di fference in
the x-coordinates of their centers and the diff erence in the y-coordinates of their centers are both
atmost 6, regardless of any dragons in between. For example, a dragon centered at (4, 5) can fire at
a dragon centered at the origin, but a dragon centered at (7, 0) cannot. A dragon cannot fi re at
itself. What is the maximum number of dragons that can fire at a single dragon simultaneously?

16 programers are playing

16 programers are playing in a single elimination tournament. Each player has a diff erent skill level
and when two play against each other the one with the higher skill level will always win. Each round,
each programer plays a match against another and the loser is eliminated. This continues until only
one remains. How many diff erent programers can reach the round that has 2 players remaining?

positive integers

Find all positive integers n for which 3n − 4, 4n − 5, and 5n − 3 are all prime.

the greatest 9-digit number

Find the greatest 9-digit number whose digits’ product is 9!.

three squares with side lengths 2, 3, and 6

Given three squares with side lengths 2, 3, and 6, cut two of them and reassemble the resulting
five pieces into a square of side length 7. (By cut we mean dissect the square into two pieces
by a polygonal line.)

246, 462, and 624

The numbers 246, 462, and 624 are all divisible by 6. What is the greatest possible common
divisor of the three-digit numbers abc, bca, and cab when a, b, and c are all different?

The harmonic mean of

The harmonic mean of two positive integers is 2006. Find the greatest possible value of their
arithmetic mean.

an equiangular hexagon

Is there an equiangular hexagon whose side lengths are (in some order) 2006, 2007, 2008, 2009,
2010, and 2011?

A company reports annually

A company reports annually. It has been noted that the company recorded a profit over every
period of m consecutive years and a loss over every period of n consecutive years.
Find (in terms of p and q) the maximum possible length of time the company has been in business

PIXIE and VEDANSI play a triangle game

PIXIE and VEDANSI play a triangle game. PIXIE fi rst draws a triangle ABC with area 1, and
VEDANSI picks a point X inside the triangle ABC. pixie then draws segments DG, EH, and FI,
all through X, such that D and E are on BC, F and G are on AC, and H and I are on AB. The ten
points must all be distinct. Finally, let S be the sum of the areas of triangles DEX, FGX, and HIX. VEDANSI earns S points, and PIXIE earns (1 - S) points. If both players play optimally to maximize
the amount of points they get, who will win and by how much?

ABCD be a quadrilateral

Let ABCD be a quadrilateral with AB // CD, AB = 16 units, CD = 12 units, and BC < AD.
A circle with diameter 12 units is inside of ABCD and tangent to all four sides. Find BC.

A rod is broken

A rod is broken into three pieces; the two break points are chosen at random.
What is the probability that the three pieces can be joined at the ends to form a triangle? 

A square of

A square of 10 units side having a circle of 1 cm diameter whose centre is the centre of the square.
A coin of diameter 1 cm is tossed onto the square. Find the probability that this tossed coin intersects the circle. 

A random point A

A random point P is uniformly distributed in a square of side length 1 unit.
Find the probability of the following events:-

• the distance from P to a fixed side of the square does not exceed x, (x < 1)
• the distance from P to the nearest side does not exceed x, ( x < 1/2)
• the distance from P to the centre does not exceed x, { x < 1/ sqrt. (2)}
• the distance from P to a fixed vertex of the square does not exceed x, ( x < or = 1/ sqrt. 2)

P is a randomly chosen

P is a randomly chosen point in a unit square. Connect P to the vertices of the base. Find the probability that the triangle is acute angled; What will be the probability for a right angled triangle. 

all pairs (m, n)

 Find all pairs (m, n) of positive integers such that m(n + 1) + n(m − 1) = 2013

the system

• a!b! = 6!
• b!c! = 7!
• c!a! = 10!

find the triplet a, b, c

k and l are positive integers

If k and l are positive integers such that k divides l, show that for every positive integer m,
1 +(k + m)l and 1+  ml are relatively prime.

relatively prime

Two integers m and n are called relatively prime if (m,n) = 1. Prove that among any five consecutive positive integers there is one integer which is relatively prime to the other four integers.  

mn = 25!

Find the number of rational numbers (m/n), where m,n are relatively prime positive integers satisfying m < n and mn = 25!

13 is the required prime

suppose p is a prime number such that (p - 1)/4 and (p + 1)/2 are also primes.

Show that p = 13.

a prime number

Show that if a prime number p is divided by 30, then the remainder is either a prime or is 1.

111...........1

Show that the number (11................1) with 729 digits is divisible by 729

two quadratic polynomials

Let f(x) and g(x) be two quadratic polynomials all of whose coefficients are rational numbers. Suppose f(x) & g(x) have a common irrational roots. Show that g(x) = r.f(x) for some rational number r. 

three perfect squares

If n is a positive integer greater than 1 such that (3n + 1) is perfect square, then show that (n + 1) is the sum of  the three perfect squares.

100000000

How many natural number less than 100000000 are there, whose sum of digits equals to 7? 

real numbers a, b , c

Let a, b, c be real numbers satisfying a < b < c, a + b + c = 6 and ab + bc + ca = 9.
Prove that 0 < a < 1 < b < 3 < c < 4.

P (x) be a polynomial with real coefficients

Let P (x) be a polynomial with real coefficients.
Show that there exists a nonzero polynomial Q (x) with real coefficients such that
P (x) Q (x) has terms that are all of a degree divisible by 10^9 .

polynomial o f degree 3n

P (x) is a polynomial o f degree 3n such that
P(0) = P ( 3 ) = ..... = P(3n) = 2;
P ( I ) = P ( 4 ) = .............. = P ( 3 n - 2) = 1;
P ( 2 ) = P ( 5 ) = .................. = P ( 3n - l ) = 0;
and   P(3n + I ) = 730.
Determine n.

P(n + 1 )

If P(x) denotes a polynomial of degree n such that P(k) = k/(k + 1 ) for k = 0, 1 , 2, . . . , n,
determine P(n + 1 ) .

the number o f partitions

Show that for each positive integer n, the number o f partitions o f n into unequal parts is equal to the number of partitions of n into odd parts.
For example, if n = 6, there are four partitions into unequal parts, namely
                               1 + 5, 1 + 2 + 3, 2+4, 6.
And there are also four partitions into odd parts,
                                                   1 + 1 + 1 + 1 + 1 + 1, 1 + 1 + 1 + 3, 1 +5, 3 + 3.

distinct ordered pairs

How many distinct ordered pairs (a, b) of non-negative integers satisfy 2 a + 5 b = 100?

a circle of n lights

Given a circle of n lights, exactly one of which is initially on, it is permitted to change the state of a bulb provided one also changes the state of every dth bulb after it (where d is a a divisor of n strictly less than n), provided that all n/d bulbs were originally in the same state as one another.
For what values of n is it possible to tum all the bulbs on by making a sequence of moves of this kind?

66 x 62

Is is possible to tile a 66 x 62 rectangle with 12 x 1 rectangles?

A bubble chamber

A bubble chamber contains three types of subatomic particles: 10 particles o f type X, 11 o f type Y,

111 o f type Z . Whenever an X- and Y-particle collide, they both become Z-particles. Likewise,

Y- and Z-particles collide and become X particles and X- and Z-particles become Y-particles upon collision. Can the particles in the bubble chamber evolve so that only one type is present?

integral length

A rectangle is tiled with smaller rectangles, each of which has at least one side of integral length.
Prove that the tiled rectangle also must have at least one side of integral length.

an odd number of games

If 1 27 people play in a singles tennis tournament, prove that at the end of tournament,
the number of people who have played an odd number of games is even.

Invertible

Let A and B be 2 x 2 matrices with integer entries such that A , A + B , A + 2B , A + 3B, and A + 4B are all invertible matrices whose inverses have integer entries.
Show that A + 5B is invertible and that its inverse has integer entries

many zeroes

• How many zeroes does 6250! end with?
• If n! has exactly 20 zeroes at the end, find n. how many such n are there?
• find n! that ends with exactly 497 zeroes.

base 11 or 9

A three digit number in base 11, when expressed in base 9, has its digits reversed.
Find the number.

360

find the smallest integer with exactly 24 divisors

regular 9-gon

Let each of the vertices of a regular 9-gon be colored black or red.
(i) Show that there are two adjacent vertices of the same color.
(ii) Show that there are three adjacent vertices of the same color forming an isosceles triangle.  

Minima

If a , b, c be a positive proper fraction such that a + b + c = 2; then prove that abc greater than or equals to
eight times of (1 - a) (1 - b) (1 - c)

8n + 4

Show that, for any positive integer n, the sum of  (8n + 4) consecutive positive integers cannot be a perfect square

Arithmetic Sequences

Show that there cannot exist three prime numbers, each greater than 3, which are in A.P. with a common difference less than 5.

Let k > 3 be an integer. Show that it is not possible for k prime numbers, each greater than k, to be in A.P. with a common difference less than or equal to (k + 1).

complex number

Let z be a complex number such that z & (z + 1) have modulus 1. If for a positive integer n, (z + 1) be an nth. root of unity, then show that z is an nth. root of unity and n is a multiple of 6.

Parallel to AO

Let ABC be any triangle and let O be a point on the line segment BC. Show that there exists a line parallel to AO which divides the triangle ABC into two equal parts of equal area.

r = s

Let R and S be two cubes with sides of length r and s respectively, where r and s are natural numbers.
Show that the difference of their volumes equals to the difference of their surface areas, 
if and only if r = s. 

Natural Number

Let m be a natural number with digits consisting entirely of 6's and 0's.
Prove that m is not the square of a natural number.

20096

Write the set of all positive integers in triangular array as

1           3         6           10          15       ...

2           5         9           14          ...       ...

4           8        13           ...          ...       ...

7          12        ...           ...          ...       ...

11        ...         ...          ...          ...       ...

Find the row number and column number where 20096 occurs. 

For example 13 appears in the 3rd. Row and 3rd. Column.

M X N

If each of the m points in one straight line be joined to each of the n points by straight lines terminated then excluding the given points, find the number of points where the line will intersect.  

Six Digit Natural Numbers

Find the number of 6 digit natural numbers where each digit appears at least twice. 

Difference of Numbers

Divide the numbers 1, 2, 3, 4, 5 into two arbitrarily chosen sets. Prove that one of the sets contains two numbers and their difference.

Lewis Carroll

Lewis Carroll, the famous author of Alice in Wonderland, Through the Looking Glass, The Hunting of the Shark and other wonderful works, was a mathematician whose real name was Charles Lutwidge Dodgson (1832-1898). Here is a problem from his book " A Tangled Tale".
Let S be the set of prisoners, E be the set of those that lost an Eye, H be the set that lost an Ear, A those that lost an Arm and L those that lost a Leg.
Given that n(E) = 70%, n(H) < 75% , n(A) = 80% and n(L) = 85%, find what % at least must have lost all four? 

Tossed Coin

A square of 10 cms side has a circle of 1 cm diameter, whose centre is the centre of the square. A coin of diameter 1 cm is tossed onto the square. Find the probability that this tossed coin intersects the circle.

Green plus Red not a Prime

There are 6 red balls and 8 green balls in a bag. Five balls are drawn out at random and placed in a red box. The remaining 9 balls are put in a green box. What is the probability that the number of red balls in the green box plus the number of green balls in the red box is not a prime number?

A Square Array of Dots

Consider a square array of dots, colored red or blue, with 20 rows and 20 columns. Whenever two dots of the same color are adjacent in the same row or column; they are joined by a segment of their common color. Adjacent dots of unlike colors are joined by a black segment. There are 219 red dots, 39 of them on the border of the array, not at the corners. There are 237 black segments. How many blue segments are there?

five mathematicians

During a certain lecture each of five mathematicians fell asleep exactly twice. For each pair of these mathematicians, there was some moment when both were sleeping simultaneously. Prove that at some moment, some three of them were sleeping simultaneously.

colour problem

Ball and Colour problem: There are certain number of balls and they are painted with the following conditions:

• Every two colours appear on exactly one ball.
• Every two balls have exactly one colour in common.
• There are four colours such that any three of them appear on one ball.
• Each ball has three colours.

Find the number of balls and the number of colours used.

ARTILLERY TARGET

An artillery target may be either at point A with probability 8/9 or at point B with probability 1/9, we have 21 shells, each of which can be fired either at point A or at point B. Each shell may hit target independently of the other shells, with probability 1/2. How many shells must be fired at point A to hit the target with maximum probability? 

Greatest Sequences

Pixie and Vedansi together thought of ten quadratic trinomials. Then,
Vedansi began calling consecutive natural numbers starting with some
natural number. After each called number, Pixie chose one of the ten
polynomials at random and plugged in the called number. The results
were recorded on the board. They eventually form a sequence.
After they fi nished, their sequence was arithmetic.
What is the greatest number of numbers that Vedansi could have called out?

Locus

Let A & B be two fixed points on a fixed straight line. 

Two circles touch this line at A & B respectively and tangent to each other at M. 

When the circles vary, the locus of M forms

(A) A circle with diameter AB minus points A & B.

(B) An ellipse with major axis AB minus points A & B.

(C) An ellipse with minor axis AB minus points A & B.

(D) A square with diagonal AB minus points A & B

Symmetric Relation

Find the probability that a randomly chosen relation from a set A = { 1,2,3,.....,n}  to itself is a symmetric relation

Affirmative

A smart cafe conducted a survey by asking every fourth person entering their store if they already a smart

tablet. On a given day out of 100 total respondents, 80 answered affirmative. Next day the same survey was

conducted by choosing every fifth person entering the store and the number of respondents was again 100. 

Which of the following is the most likely number of respondents answering in the affirmative? 

(a) 64                  (b) 78           (c) 100                   (d) 92

Pairwise Different

101 distinct numbers are chosen among the integers between 0 and 1000. Prove that, among the absolute values of their pairwise di fferences, there are ten di fferent numbers not exceeding 100.

2013

On each of the cards written in 2013 by number, all of these 2013 numbers are diff erent. The cards are turned down by numbers. In a single move is allowed to point out the ten cards and in return will report one of the numbers written on them (do not know what). For what most W guaranteed to be able to fi nd W cards for which we know what numbers are written on each of them?

Monic poly

Let P(x) and Q(x) be (monic) polynomials with real coe fficients (the fi rst coeffi cient being equal to 1), and
deg P(x) = deg Q(x) = 10. Prove that if the equation P(x) = Q(x) has no real solutions, 
then P(x + 1) = Q(x - 1) has a real solution.

Real Solutions

Given three distinct real numbers a, b, and c, show that at least two of the three following
equations: (x - a)(x - b) = x - c;  (x - c)(x - b) = x - a; (x - c)(x - a) = x - b
have real solutions.

Least Perimeter

In triangle ABC, AB = AC and the lengths of the sides are integers. Let E be the mid point of BC. If the length of BE, AE & AB are in A.P., then find the least possible perimeter of triangle ABC. 

Unconventional

An unconventional die having face value of 1,0,-1,2,3 & -2 thrown thrice. find the probability that the sum of the outcome will (i) Zero & (ii) Six